Anisotropic compact stars: Constraining model parameters to account for physical features of tidal Love numbers
Shyam Das, Saibal Ray, Maxim Khlopov, K.K. Nandi, Bikram Keshari Parida
aa r X i v : . [ g r- q c ] F e b MNRAS , ?? – ?? (2021) Preprint 16 February 2021 Compiled using MNRAS L A TEX style file v3.0
Anisotropic compact stars: Constraining model parameters to account forphysical features of tidal Love numbers
Shyam Das ⋆ , Saibal Ray † , Maxim Khlopov ‡ , K.K. Nandi § , B.K. Parida ¶ Department of Physics, P. D. Women’s College, Jalpaiguri 735101, West Bengal, India Department of Physics, Government College of Engineering and Ceramic Technology, Kolkata 700010, West Bengal, India National Research Nuclear University, MEPHI (Moscow Engineering Physics Institute), Moscow 115409, Russia,CNRS, Astroparticule et Cosmologie, Universit´e de Paris, F-75013 Paris, France & Institute of Physics, Southern Federal University,344090 Rostov on Don, Russia Bashkir State Pedagogical University & Zel’dovich International Centre for Astrophysics, Ufa 450000 (RB), Russia Department of Physics, Pondicherry University, Kalapet, Puducherry 605014, India
Accepted . Received ; in original form
ABSTRACT
In this paper, we develop a new class of models for a compact star with anisotropic stresses inside the matter distribution. Byassuming a linear equation of state for the anisotropic matter composition of the star we solve the Einstein field equations. Inour approach, for the interior solutions we use a particular form of the ansatz for the metric function g rr . The exterior solutionis assumed as Schwarzschild metric and is joined with the interior metric obtained across the boundary of the star. Thesematching of the metrices along with the condition of the vanishing radial pressure at the boundary lead us to determine themodel parameters. The physical acceptability of the solutions has verified by making use of the current estimated data availablefrom the pulsar U − . Thereafter, assuming anisotropy due to tidal effects we calculate the Love numbers from our modeland compare the results with the observed compact stars, viz. KS − , U − , U − , U − , SAX J . − and EXO − . The overall situation confirms physical viability of the proposed approach, whichcan shed new light on the interior of the compact relativistic objects. Key words: stars: neutron - pulsars: general - equation of state - gravitation - hydrodynamics - methods: analytical.
In General Relativity (GR), while solving the problems with com-pact stars, it is a general custom to assume the objects with fea-tures of spherically symmetric and isotropic nature. However, theisotropy and homogeneity of astrophysical compact stellar objectsideally may have solvable features but they need not be general phys-ical characteristics of the stellar objects. As such the fluid pressuremay have two distinct components which are responsible to provideanisotropic factor ( ∆ = p t − p r ) where inhomogeneity results dueto the radial pressure ( p r ) and tangential pressure ( p t ) and hencemay make the internal system of matter distribution devoid of anidealized isotropic case. This idea was first discussed by Ruderman(1972) in his extensive review work on pulsar’s structure and dy-namics. Later on several scientists (Canuto 1974; Bowers & Liang1974; Herrera & Santos 1997) highlighted the issue in their works.However, different factors are thought to be responsible for thisanisotropy, namely (i) very high density region in the core region,(ii) various condensate states (like pion condensates, meson con-densates etc.), (iii) superfluid 3A, (iv) mixture of fluids of differ- ⋆ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] § E-mail: [email protected] ¶ E-mail: [email protected] ent types, (v) rotational motion, (vi) presence of the magnetic field,(vii) phase transition, (viii) relativistic particles in the compact starsetc. (Ivanov 2002; Schunck & Mielke 2003; Mak & Harko 2003;Varela et al. 2010; Rahaman et al. 2010, 2011, 2012a,b; Kalam et al.2012; Deb et al. 2015; Shee et al. 2016; Maurya et al. 2016, 2017;Deb et al. 2017; Maurya et al. 2018).However, there is another factor regarding anisotropy in the com-pact stars that has been in speculation in the form of gravitationaltidal effects. This is thought to be responsible for deformation andhence anisotropy in the fluid distribution in the stars (Hinderer2008; Doneva & Yazadjiev 2012; Herrera & Barreto 2013;Biswas & Bose 2019; Rahmansyah et al. 2020; Roupas & Nashed2020; Bhar, Das & Parida 2020; Das, Parida & Sharma 2020;Chatziioannou 2020). In the present work we will emphasize onthis physical ingredient of anisotropic nature of compact stars andtherefore shall involved in calculating the Love numbers arising dueto tidal effect. However, though the equilibrium configuration of theneutron star may get tidally deformed by developing a multipolarstructure, in our work for the sack of simplicity, shall considerquadrupole moment instead of multipole moment.It is argued by Pretel (2020) that the equation of state (EOS)plays a fundamental role in determining the internal structure ofsuch stars and, consequently, in imposing stability limits. As faras the LIGO-Virgo constraints on the EOS for nuclear matter asa result of observation of the event GW170817 is concerned it © 2021 The Authors
Das, Ray, Khlopov, Nandi & Parida is more crucial Pretel (2020); Chatziioannou (2020). Accordingto Fattoyev et al. (2020) GW170817 has provided stringent con-straints on the EOS of neutron rich matter at a few times nucleardensities from the determination of the tidal deformability of a M = 1 . M ⊙ neutron star (Bauswein et al. 2017; Abbott et al.2018; Fattoyev, Piekarewicz & Horowitz 2018; Annala et al. 2018;Most et al. 2018; Tews, Margueron & Reddy 2018; Malik et al.2018; Radice et al. 2018; Tews, Margueron & Reddy 2018, 2019;Capano et al. 2019). Therefore, a realistic EOS is always in demandfor exploring the effects it can have on the physical characteristics ofa desirable stable anisotropic compact stars. In the present investiga-tion we are employing a linear EOS of the form p r = αρ + β where α and β are constants so that there are ample opportunity to tune itwith the observational situation for various astrophysical system ofdense objects.The paper is organised as follows. In Section 2 we present the as-sociated Einstein field equations describing a spherically symmetricstatic anisotropic configuration. By assuming a particular form forthe g rr metric potential and a linear EOS, we have solved the sys-tem in Section 3. The matching conditions required for the smoothconnection of the interior space-time to the vacuum Schwarzschildexterior are provided in Section 4. In Section 5 we provide the boundon the model parameters required for the physical analysis of our so-lution. The physical viability of our model is studied in Section 6along with the tidal deformation and Love numbers in Section 7. Weconclude with a discussion in Section 8. We write the line element describing the interior space-time of aspherically symmetric star in standard coordinates x = t , x = r , x = θ , x = φ as ds = − e ν ( r ) ( r ) dt + e λ ( r ) dr + r ( dθ + sin θdφ ) , (1)where, e ν ( r ) and e λ ( r ) are the gravitational potential are yet to bedetermined.We assume that the matter distribution of the stellar interior isanisotropic in nature and described by an energy-momentum tensorof the form T αβ = ( ρ + p t ) u i u j + p t g ij + ( p r − p t ) χ i χ j , (2)where ρ represents the energy-density, p r and p t , respectively denotefluid pressures along the radial and transverse directions, u i is the -velocity of the fluid and χ i is a unit space-like -vector along theradial direction so that u i u i = − , χ i χ j = 1 and u i χ j = 0 .The Einstein field equations for the line element (1) are obtainedas (in the geometrized system of units having G = c = 1 ) π ρ = (cid:0) − e − λ (cid:1) r + λ ′ e − λ r , (3) π p r = ν ′ e − λ r − (cid:0) − e − λ (cid:1) r , (4) π p t = e − λ (cid:18) ν ′′ + ν ′ − ν ′ λ ′ + 2 ν ′ r − λ ′ r (cid:19) , (5)where primes ( ′ ) represent differentiation with respect to the radialcoordinate r .Making use of Eqs. (4) and (5), we define the anisotropic parame- ter of the stellar system as π ∆( r ) = ( p t − p r )= e − λ (cid:18) ν ′′ + ν ′ − ν ′ λ ′ − r ( ν ′ + λ ′ ) + 4 r ( e λ − (cid:19) . (6)The anisotropic force which is defined as r will be repulsive orattractive in nature depending upon whether p t > p r or p t < p r .Thus we have a system of four equations Eq. (3)-Eq. (6) with independent variables, namely e λ , e ν , ρ , p r , p t and ∆ . We need tospecify two of them to solve the system. In this model we solve thesystem by assuming a particular metric anasatz g rr and the interiormatter distribution to follow a linear equation of state. To develop a physically reasonable model of the stellar configura-tion, we assume that the metric potential g rr is given by e λ ( r ) = 1 + ar + br , (7)where a and b are the constants to be determined from the matchingconditions. This metric potential was earlier proposed by Tolman(1939) to model realistic compact stellar object. Interestingly, we areusing the same metric component to describe relativistic anisotropicstellar objects with a prescribed linear equation of state of the form p r = αρ + β, (8)where α and β are constants.Substituting (7) in Eq. (3) and using Eq. (8) and Eq. (4), we have ν ′ = 1 ar + br + 1 (cid:2) r (cid:0) a αr + a r + 2 abαr + 8 πβ (cid:0) ar + br + 1 (cid:1) + 2 abr + 3 aα + a + b αr + b r + 5 bαr + br (cid:1)(cid:3) . (9)Integrating we have, ν = 112 r (cid:2) α + 1) (cid:0) a + br (cid:1) + 8 πβ (cid:0) ar + 2 br + 6 (cid:1)(cid:3) + α log (cid:0) ar + br + 1 (cid:1) + c , (10)and hence e ν ( r ) = A (cid:0) ar + br + 1 (cid:1) α × exp (cid:20) r (cid:0) α + 1) (cid:0) a + br (cid:1) + 8 πβ (cid:0) ar + 2 br + 6 (cid:1)(cid:1)(cid:21) , (11)where A = e c is a constant of integration. Consequently, the physi- MNRAS , ?? – ?? (2021) nisotropic compact stars cal quantities are obtained as ρ = r (cid:0) a + 5 b (cid:1) + 2 ab r + 3 a + b r π (1 + a r + b r ) , (12) p r = α (cid:0) r (cid:0) a + 5 b (cid:1) + 2 ab r + 3 a + b r (cid:1) π (1 + a r + b r ) + β, (13) p t =2 πβ r (cid:0) ar + br + 1 (cid:1) + 132 π (cid:2) a ( α + 1) + b ( α + 1) r + 4( α − αr (cid:0) a − b (cid:1) ( ar + br + 1) + a (cid:0) − α + 6 α − (cid:1) + 4 b ( α (2 α − − r ( ar + br + 1) + ( α + 1) (cid:0) aα + a + b (7 α + 3) r (cid:1) ar + br + 1 + 12(1 + ar + br ) (cid:2) β (cid:0) a ( α + 1) r + 2 ar (cid:0) b ( α + 1) r + 3 α + 4 (cid:1) + b ( α + 1) r (cid:0) br + 5 (cid:1)(cid:1)(cid:3) , (14) ∆ = 132 (cid:2) βr ( aα + a + 4 πβ ) + 64 αβ + 32 β + 16 βr (4 πaβ + bα + b ) + 64 πbβ r + 4( α − αr (cid:0) a − b (cid:1) π ( ar + br + 1) + ( α + 1) (cid:0) a + br (cid:1) π + 4 b ( α (2 α − − r − a (cid:0) α + α + 1 (cid:1) π ( ar + br + 1) + 1 π ( ar + br + 1) [ − πα + π ) βa (cid:0) α − π (2 α + 1) βr + 1 (cid:1) + b ( α (7 α + 6) + 3) r (cid:3)(cid:3) (15)The parameter β can be expressed as β = − αρ R , where R is theradius of the star and ρ R is the surface density given by ρ R = R (cid:0) a + 5 b (cid:1) + 2 ab R + 3 a + b R π (1 + a R + b R ) . (16)This ensures that radial pressure p r ( r = R ) = 0 . The central density ρ ( r = 0) can be obtained from Eq. (12) as ρ c = 3 a / π, (17)which shows that we must have a > .The anisotropy vanishes at the centre i.e., ∆( r = 0) = 0 .The mass contained within a sphere of radius r is defined as m ( r ) = 12 r Z ω ρ ( ω ) dω, (18)which on integration yields m ( r ) = 116 π (cid:20) r − rar + br + 1 (cid:21) , (19)obviously, m ( r = 0) = 0 . We need to match the interior solution to the Schwarzschild exterior ds = − (cid:18) − Mr (cid:19) dt + (cid:18) − Mr (cid:19) − dr + r Ω , (20) across the boundary R where M = m ( R ) is the total mass and Ω = dθ + sin θ dφ .The matching conditions determine the constants as a = bR − M (cid:0) bR + 1 (cid:1) R (2 M − R ) , (21) b = 14 α (cid:18) αM − αMR + 3(5 α + 1) R R ( R − M ) − s R ( R − M ) ξ ! , (22) β = 116 πR (cid:2) α (cid:0) − M + 30 MR − R (cid:1) + R − R ( − M + R ) s R ( R − M ) ξ ! , (23)where, ξ =256 α M − α M R + 12 α (139 α + 32) M R − α (89 α + 29) MR + 9(5 α + 1) R − αR ( R − M ) (cid:20) log( A ) + α log (cid:18) RR − M (cid:19) − log (cid:18) − MR (cid:19)(cid:21) . For a physically acceptable stellar model, it is reasonable to assumethat the following conditions should be satisfied (Delgaty & Lake1998): (i) ρ > , p r > , p t > ; (ii) ρ ′ < , p ′ r < , p ′ t < ; (iii) dp r dρ ; dp t dρ and (iv) ρ + p r + 2 p t > . In addition,it is expected that the solution should be regular and well-behavedat all interior points of the stellar configuration. Based on the aboverequirements, bounds on the model parameters are obtained in thissection.(i) Regularity Conditions: (a) The metric potentials e λ ( r ) > , e ν ( r ) > for r R. For appropriate choice of the model parameters, the above re-quirements are fulfilled in our model. The gravitational potentialsin this model satisfy, e ν (0) = A = constant, e λ (0) = 1 , i.e., finiteat the center ( r = 0 ) of the stellar configuration. Also one caneasily check that ( e ν ( r ) ) ′ r =0 = ( e λ ( r ) ) ′ r =0 = 0 . These imply thatthe metric is regular at the center and well behaved throughoutthe stellar interior which will be shown graphically.(b) ρ ( r ) > , p r ( r ) > , p t ( r ) > for r R .From Eq. (12), we note that density remains positive if a > .Equation (13) shows that since dp r /dρ ( r = 0) = α is the soundspeed must be between and , so < α < . From equation(14), we have p t ( r = 0) = 132 π (cid:2) a (cid:0) − α + 6 α − (cid:1) + a ( α + 1) +( α + 1)(3 aα + a )] + β. (24)We note that for the tangential pressure remain positive thecentre r = 0 . Fulfillment of the requirements throughout the star MNRAS , ?? – ?? (2021) Das, Ray, Khlopov, Nandi & Parida e γ e λ ( km ) M e t r i ces
4U 1608 - Figure 1.
The metric potentials e ν and e λ are plotted against the radial coor-dinate r inside the stellar interior. can be shown by graphical representation.(c) p r ( r = R ) = 0 .From Eq. (13), we note that the radial pressure vanishes at theboundary R if we set β = − αρ R , where ρ R is the surface density.In this context we would like to emphasis on the point that if wechoose ρ ( r = R ) = 0 then Eq. (8) becomes p r ( r = R ) = β = 0 (because at boundary the radial pressure is zero). Also, as α and β are constant parameters for a particular star, so they should remainsame always. This suggests that we should not make ρ ( r = R ) =0 at the boundary, however this may be considered as a specialcase.(ii) Causality Condition:
The causality condition demands that dp r dρ ; dp t dρ at all interior points of the star. Let usnow represent the related expressions as follows: dp r dρ = α, (25) dp t dρ = − (cid:0) a r + b r (cid:1) B + B + B ) × " − α − αr (cid:0) a − b (cid:1) (cid:0) a + 2 b r (cid:1) (1 + a r + b r ) + 4 α ( α − (cid:0) a − b (cid:1) (1 + a r + b r ) − (cid:0) a + 2 b r (cid:1) (cid:0) B + B r (cid:1) (1 + a r + b r ) +4 (cid:0) a + 2 b r (cid:1) (cid:0) B + B r (cid:1) (1 + a r + b r ) + 2 βr ( a β + 2 b ( α + 1))+ b ( α + 1)(7 α + 3) − a (2 α + 1) β a r + b r + 4 b ( α (2 α − − a r + b r ) + β (2 a ( α + 1) + β )+ b ( α + 1) + 3 b β r (cid:3) , (26)where, B = a r + a (cid:0) b r + 5 (cid:1) , B = ab r (cid:0) b r + 13 (cid:1) , B = + b (cid:0) b r + 12 b r − (cid:1) , B = 3 a α + 4 a α + a − αβ − β , B = ( b ( α + 1)(7 α + 3) − a (2 α + 1) β ) , B = a (2 α − α + 1) and B = 2 b (cid:0) − α + 5 α + 1 (cid:1) . At the centre r = 0 , dp t dρ > i.e., dp t dρ = − a ( α − α −
1) + 2 a (3 α + 2) β + 40 b α + β
20 ( a − b ) > , (27)Also according to Zeldovich’s condition Zel’dovich (1962, 1972), p r /ρ must be at the center. Therefore, (3 aα + β )3 a .(iii) Energy Condition:
For an anisotropic fluid sphere for beingphysically accepted matter composition, all the energy conditions,namely Weak Energy Condition (WEC), Null Energy Condition(NEC), Strong Energy Condition (SEC) and Dominant Energy Con-dition (DEC) are satisfied if and only if the following inequalitieshold simultaneously in every point inside the fluid sphere.NEC : ρ + p r > ; ρ + p t > ; ,WEC : p r + ρ > , ρ > ,SEC : ρ + p r > , ρ + p r + 2 p t > ,DEC : ρ > | p r | , ρ > | p t | .We have from SEC ρ + p r + 2 p t ( r = 0) = 3(3 aα + a + 8 πβ )8 π > , ⇒ β > − a (1 + 3 α )8 π (28)(iv) Monotonic decrease of density and pressures:
A realisticstellar model should have the following properties: dρdr , dp r dr , dp t dr for r R .Now, dρdr = − r ( C + C + C )8 π (1 + a r + b r ) , (29) dp r dr = − αr ( C + C + C )8 π (1 + a r + b r ) , (30) dp t dr = 116 π r (cid:20) − C (1 + a r + b r ) + C (1 + a r + b r ) + β (2 a ( α + 1) + β ) − C ( C + C )(1 + a r + b r ) +4 C ( C + C )(1 + a r + b r ) + 2 βr ( a β + 2 b ( α + 1))+ b ( α + 1)(7 α + 3) − a (2 α + 1) β a r + b r + 4 b ( α (2 α − − a r + b r ) + b ( α + 1) + 3 b β r (cid:21) , (31)where C = a r + a (3 b r + 5) , C = ab r (cid:0) b r + 13 (cid:1) , C = b (cid:0) b r + 12 b r − (cid:1) , C = 12 C ( α − αr (cid:0) a − b (cid:1) , C = (cid:0) a + 2 b r (cid:1) , C = 4( α − α (cid:0) a − b (cid:1) , C = r ( b ( α +1)(7 α + 3) − a (2 α + 1) β ) , C = 3 a α + 4 a α + a − αβ − β , C = a (cid:0) α − α + 1 (cid:1) , C = 2 b (cid:0) − α + 5 α + 1 (cid:1) r .With the choices of the model parameters within their properbound it can be shown that both density and radial pressure decreaseradially outward. MNRAS , ?? – ?? (2021) nisotropic compact stars ρ ' p r ' p r ' - - - - - -
200 r ( km ) G r a d i e n t
4U 1608 - Figure 2.
Variations of the density and pressure gradients. ( km ) ρ ( M e V / f m )
4U 1608 - Figure 3.
Fall-off behaviour of the energy density. p r p t ( km ) p r es u r es ( M e V / f m )
4U 1608 - Figure 4.
Fall-off behaviour of the pressure. ( km ) Δ ( M e V / f m )
4U 1608 - Figure 5.
Radial variation of anisotropy. ( km ) M(cid:0)(cid:1)(cid:2) ( M ⊙ )
4U 1608 - Figure 6.
Verification of the stellar mass. ρ + p r + p t ρ + p r ρ + p t ( km ) E n e r g yc ond i t i on s ( M e V f m - )
4U 1608 - Figure 7.
Verification of the energy condition. e - λ e γ - MR r ( k(cid:22) ) M e t r i ces
4U 1608 - Figure 8.
Matching of the metrices at the boundary.
A star remain in static equilibrium under the forces namely, grav-itational force ( F g ), hydrostatics force ( F h ) and anisotropic force( F a ). This condition is formulated mathematically as TOV equationby Tolman-Oppenheimer-Volkoff which is described by the conser-vation equation given by ∇ µ T µν = 0 , (32)known as TOV equation.Now using the expression given in (2) into (32) one can obtain thefollowing equation: − ν ′ ρ + p r ) + 2 r ( p t − p r ) = dp r dr . (33)The Eq. (33) can be written as, F g + F h + F a = 0 , (34) MNRAS , ?? – ?? (2021) Das, Ray, Khlopov, Nandi & Parida f g f h f a - - - ( (cid:23)(cid:24) ) F(cid:25)(cid:26)(cid:27)(cid:28)(cid:29)
4U 1608 - Figure 9.
Variation of the forces. where the expression for F g , F h and F a are obtained as: F g = r π ( ar + br + 1) (cid:2) a r (cid:0) α + 8 πβr + 1 (cid:1) + 8 πβ (cid:0) br + 1 (cid:1) + br (cid:0) b ( α + 1) r + 5 α + 1 (cid:1) + a (cid:0) b ( α + 1) r + 16 πbβr + 3 α + 16 πβr + 1 (cid:1)(cid:3) × h a r (cid:0) α + 8 πβr + 1 (cid:1) + 8 πβ (cid:0) br + 1 (cid:1) + b ( α + 1) r (cid:0) br + 5 (cid:1) + a (3( α + 1)2 b ( α + 1) r + 16 πbβr + +16 πβr (cid:1)(cid:3) , (35) F h = 14 π ( ar + br + 1) αr (cid:2) a − b + b r + ar (cid:0) a + 13 b (cid:1) + 3 br (cid:0) a + 4 b (cid:1) + 3 ab r (cid:3) (36) F a = r (cid:20) β ( aα + a + 4 πβ ) + b ( α + 1) π + 16 βr (4 πaβ + bα + b ) + 64 πbβ r + 4 χ πη + 2 χ πη + χ πη (cid:21) (37)Where, η = ar + br + 1 , χ = ( α − α (cid:0) a − b (cid:1) , χ = a (cid:0) α + α + 1 (cid:1) + ab (cid:0) α + α + 1 (cid:1) r + 2 b ( α (2 α − − , χ = a ( α + 1) + ab ( α + 1) r + 16 πa (2 α + 1) β + b (cid:0) α + 32 π (2 α + 1) βr + 6 α + 3 (cid:1) .The three different forces are plotted in Fig. 9 for the com-pact stars U − . The figure shows that hydrostatics andanisotropic forces are positive and is dominated by the gravitationalforce which is negative to keep the system in static equilibrium. The adiabatic index which is defined as
Γ = ρ ( r ) + p ( r ) p ( r ) dp ( r ) dρ ( r ) , (38)is related to the stability of a relativistic anisotropic stellar configu-ration.Any stellar configuration will maintain its stability if adiabaticindex Γ > / (Heintzmann & Hillebrandt 1975; Deb et al. 2019).For our solution, the adiabatic index Γ takes the value more than / throughout the interior of the compact star, as evident from Fig. 10. / Γ r Γ t ( (cid:30)(cid:31) ) Γ
4U 1608 - Figure 10.
Variation of the adiabatic index. v r !" r ( ) s6789:;<=>
4U 1608 - Figure 11.
Verification of radial sound speed. v t ?@ABCDEGHIJKLNOPQR r ( ST ) UVWXYZ[\]^
4U 1608 - Figure 12.
Verification of the transverse sound speed.
We also know that for a physically acceptable model, the velocity ofthe sound (both radial and transverse) should be less than the speedof the light, i.e., both dp r dρ , dp t dρ < which is known as the causalitycondition. The causality condition is shown to satisfy in Figs. 11and 12 .To examine the stability, we have followed the technique which isknown as “cracking method” used by Herrera (1992). Based on thismethod Abreu, Hernandez & Nunez (2007) found that for a compactstellar object, in a stable region we must have v t − v r (cid:26) < for r R ⇒ potentially stable > for r R ⇒ unstable . Figure 13 clearly indicates that for our assumed set of values theconfiguration remains stable throughout the star.
MNRAS , ?? – ?? (2021) nisotropic compact stars _‘abcdefghijlmnopqrtuvwx r ( yz ) | v t - v r |
4U 1608 - Figure 13.
Plot for | v t − v r | . {|}~(cid:127) (cid:128)(cid:129)(cid:130)(cid:131)(cid:132) (cid:133)(cid:134)(cid:135)(cid:136)(cid:137) (cid:138)(cid:139)(cid:140)(cid:141)(cid:142) (cid:143)(cid:144)(cid:145)(cid:146)(cid:147) (cid:148)(cid:149)(cid:150)(cid:151)(cid:152) (cid:153)(cid:154)(cid:155)(cid:156)(cid:157) ρ c ( km - ) d M / d ρ c ( k m )
4U 1608 - Figure 14.
Variation of dM/dρ c with respect to the central density ρ c . Depending on the mass and central density of the star, Harrison et al.(1965) and Zel’dovich & Nivokov (1971) proposed the stability con-dition for the model of compact star. From their investigation theysuggested that for stable configuration ∂M∂ρ c > , where M, ρ c de-notes the mass and central density of the compact star.For our present model ∂M∂ρ c = 3 R (3 bR + 3) . (39)Above expression of ∂M∂ρ c is positive and hence the stability conditionis well satisfied. The variation of the ∂M∂ρ c with respect to the centraldensity is depicted in Fig. 14. When a static spherically symmetric neutron star (NS) immersedin an external tidal field E ij , the equilibrium configuration of theneutron star gets tidally deformed by developing a multipolar struc-ture. However, in our calculation for the sack of simplicity, we takequadrupole moment Q ij instead of multipole moment. It is becauseof the fact that the quadrupole moment ( l = 2 ) dominates over themultiple momoent if the two binary stars are sufficiently far awayfrom each other. The relation between Q ij and E ij in the linear or-der can be written as (Hinderer 2008) Q ij = − Λ E ij . (40)Here, Λ represents the tidal deformability of the neutron star andit is related to the dimensionless tidal love number k as (Hinderer 2008) k = 32 Λ R − . (41)To determine k , consider background metric (0) g µν ( x ν ) of acompact object. With linear perturbation h µν ( x ν ) of the backgroundmetric, the the modified perturbed metric can be written as g µν ( x ν ) = (0) g µν ( x ν ) + h µν ( x ν ) . (42)We write the background geometry of the spherical static star inthe standard form (0) ds = (0) g µν dx µ dx ν = − e ν ( r ) dt + e λ ( r ) dr + r (cid:0) dθ + sin θdφ (cid:1) . (43)With this linearized metric perturbation h µν ( x ν ) , following theworks of Regge & Wheeler (1957); Biswas & Bose (2019), we re-strict ourselves to static l = 2 , m = 0 even parity perturbation.With these restriction the perturbed metric becomes h µν = diag h H ( r ) e ν , H ( r ) e λ , r K ( r ) , r sin θK ( r ) i Y m ( θ, φ ) , (44)where H , H and K are radial functions determined by perturbedEinstein equations.For the spherically static metric with anisotropic pressure,the stress-energy tensor is given as (Bowers & Liang 1974;Doneva & Yazadjiev 2012; Herrera & Barreto 2013; Pretel 2020) (0) T ξχ = ( ρ + p t ) u ξ u χ + p t g ξχ − ( p t − p r ) η ξ η χ , (45)where u χ u χ = − , η χ η χ = 1 and η χ u χ = 0 . Furthermore, theperturbed stress-energy tensor is defined as, T ξχ = (0) T ξχ + δT ξχ . Thenon-zero components of perturbed stress-energy tensor are: δT tt = − dρdp r δp r Y ( θ, φ ) , δT rr = δp r ( r ) Y ( θ, φ ) , & δT θθ = δT φφ = dp t dp r δp r ( r ) Y ( θ, φ ) . With these perturbed quantities we can write down the perturbedEinstein Field Equation becomes (where G = c = 1 ) G ξχ = 8 πT ξχ . (46)The non-zero components of the background Einstein field equa-tion gives the expressions (0) G tt = 8 π (0) T tt ⇒ λ ′ ( r ) = 8 πr e λ ( r ) ρ ( r ) − e λ ( r ) + 1 r , (47) (0) G rr = 8 π (0) T rr ⇒ ν ′ ( r ) = 8 πr p ( r ) e λ ( r ) + e λ ( r ) − r . (48)Also, we know that ∇ (0) ξ T ξχ = 0 . Choosing ξ = r , by expandingand solving the equation, we can find the expression as p ′ ( r ) = 12 r (cid:2) − p ( r ) + 4 p t ( r ) − rp ( r ) ν ′ ( r ) − rρ ( r ) ν ′ ( r ) (cid:3) . (49)The various components of perturbed part of Einstein field equa-tion (46) gives these expressions G θθ − G φφ = 0 ⇒ H ( r ) = H ( r ) = H ( r ) , (50) G θr = 0 ⇒ K ′ = H ′ + Hν ′ , (51) G θθ + G φφ = 8 π ( T θθ + T φφ ) ⇒ δp = H ( r ) e − λ ( r ) ( λ ′ ( r )+ ν ′ ( r ) ) πr dptdp . (52)Using the identity, ∂ Y ( θ, φ ) ∂θ + cot ( θ ) ∂Y ( θ, φ ) ∂θ + MNRAS , ?? – ?? (2021) Das, Ray, Khlopov, Nandi & Parida
Table 1.
Numerical values of the physical parameters in connection to different compact stars. Here the observed masses M and radii R are taken from theRef. ( ¨Ozel et al. 2016; Roupas & Nashed 2020). Compact Stars α β
A a b M ( M ⊙ ) R (km) k
4U 1608-52 0.22 -0.002 0.2896 0.011165 -0.0000191011 . +0 . − . . +1 . − . . +0 . − . . +1 . − . . +0 . − . . +1 . − . . +0 . − . . +1 . − . . +0 . − . . +1 . − . . +0 . − . +2 . − . csc ( θ ) ∂ Y ( θ, φ ) ∂φ = − Y ( θ, φ ) as well as Eqs. (47) - (52),we have the master equation for H ( r ) as − e − λ ( r ) Y ( θ,φ ) (cid:2) G tt − G rr (cid:3) = − πe − λ ( r ) Y ( θ,φ ) (cid:2) T tt − T rr (cid:3) ⇒ H ′′ ( r ) + R H ′ ( r ) + S H ( r ) = 0 , (53)where R = − (cid:20) − e λ ( r ) − r − πre λ ( r ) ( p r − ρ ( r )) (cid:21) , (54) S = − h πe λ ( r ) (cid:16) p r (cid:16) e λ ( r ) − (cid:17) − p t ( r ) − ρ ( r ) (cid:17) + 64 π r p r e λ ( r ) + 4 e λ ( r ) + e λ ( r ) + 1 r + − π dρdp r e λ ( r ) ( p r + ρ ( r )) − πe λ ( r ) ( p r + ρ ( r )) dp t dp r . (55)The expressions for e λ , ρ ( r ) , p r ( r ) , and p t ( r ) can be foundfrom the solutions of a physically acceptable model.The tidal Love number k can be calculated by matching theinternal solution with the external solution of the perturbed vari-able H ( r ) at the surface of the star (Bhar, Das & Parida 2020;Das, Parida & Sharma 2020). Then the expression of tidal love num-ber can be found in terms of y and compactness C = M/R as k = [8(1 − C ) C (2 C ( y − − y + 2)] /X, (56)where X = 5(2 C ( C (2 C ( C (2 C ( y + 1) + 3 y − − y + 13) + 3(5 y − − y + 6) + 3(1 − C ) (2 C ( y − − y + 2) log (cid:18) C − (cid:19) − − C ) (2 C ( y − − y + 2) log (cid:18) C (cid:19)(cid:19) , (57)Here C = MR and y depends on r, H and its derivatives y = rH ′ ( r ) H ( r ) (cid:12)(cid:12)(cid:12)(cid:12) r = R . (58)In order to get the numerical of k for a particular com-pactness C , let’s modify equation (53), using the eqn. (58), as(Rahmansyah et al. 2020) r y ′ + y + ( r R − y + r S = 0 . (59)Since Eq. (59) is a first order differential equation we need oneinitial condition to solve it. Setting, k = 0 as C = 0 , it is clearfrom Eq. (56) that at r = 0 , we get y ( r = 0) = 2 . Using this initialcondition along with the equation (54) & (55), one can explicitly cal-culate the solution of Eq. (59). In Fig. 16-21 the tidal Love numberis plotted against α for different compact objects. It is evident from Figure 15.
Variation of Love number k with respect to the parameter α & A for the compact star U − . (cid:158) = (cid:159)(cid:160)¡¢£ = ⁄¥ƒ§¤ = '“«‹ = ›fifl(cid:176)–†‡· (cid:181)¶•‚ „”»… ‰(cid:190)¿(cid:192)`´ˆ˜¯˘˙¨(cid:201)˚¸(cid:204)˝˛ˇ — k
4U 1820 - Figure 16.
Variation of Love number k with respect to α for the compactstar U − under the specific choice of A . (cid:209) = (cid:210)(cid:211)(cid:212)(cid:213)(cid:214)(cid:215)(cid:216) = (cid:217)(cid:218)(cid:219)(cid:220)(cid:221) = (cid:222)(cid:223)(cid:224)Æ = (cid:226)ª(cid:228)(cid:229)(cid:230)(cid:231)ŁØ Œº(cid:236)(cid:237) (cid:238)(cid:239)(cid:240)æ (cid:242)(cid:243)(cid:244)ı(cid:246)(cid:247)łøœß(cid:252)(cid:253)(cid:254)(cid:255)0(cid:0)(cid:1)(cid:2)(cid:3) (cid:4) k
4U 1608 - Figure 17.
Variation of Love number k with respect to α for the compactstar U − under the specific choice of A .MNRAS , ?? – ?? (2021) nisotropic compact stars A = (cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11) = (cid:12)(cid:13)(cid:14)(cid:15)(cid:16) = (cid:17)(cid:18)(cid:19)(cid:20) = (cid:21)(cid:22)(cid:23)(cid:24)(cid:25)(cid:26)(cid:27)(cid:28) (cid:29)(cid:30)(cid:31) !" k :;<= - >?@ Figure 18.
Variation of Love number k with respect to α for the compactstar U − under the specific choice of A . B = CDEFGHIJ = KLMNO = PQRS = TUVWXYZ[ \]^_ ‘abc defghijklmnopqrstuv w k EXO 1745 - Figure 19.
Variation of Love number k with respect to α for the compactstar EXO − under the specific choice of A . A = = = = x k yz {|}~ - Figure 20.
Variation of Love number k with respect to α for the compactstar KS − under the specific choice of A . A = = = = (cid:127) k (cid:128)(cid:129)(cid:130) J1748.9 - Figure 21.
Variation of Love number k with respect to the α for the compactstar SAX J . − under the specific choice of A . the plots that with increasing the value of α the tidal Love numberdecreases monotonically.In the Table 1, the numerical values of tidal Love number k aregiven for different compact objects with a random physically accept-able value of α and A . In this paper, we have obtained a new class of interior solutions to theEinstein field equations for an anisotropic matter distribution obey-ing a linear EOS. The solution has been shown to be regular andwell-behaved and hence could describe a relativistic compact star.The anisotropy in the present model has been assumed to be due totidal effect and hence we have calculated the Love number for sev-eral compact objects in 2 D as well as 3 D plots.Some of the salient features of the present model are as follows:(i) To show that the solution can be used as a viable model forcompact observed sources, we just consider as a specific example,the pulsar U − whose mass and radius are estimated to be M = 1 . +0 . − . M ⊙ and R = 9 . +1 . − . km, respectively ( ¨Ozel et al.2016; Roupas & Nashed 2020). For the given mass and radius, wehave determined the values of the constants a = 0 . , b = − . and A = 0 . for arbitrarily chosen values of α = 0 . , β = − . . For physical acceptability of our model,using the values of the constants and plugging the values of G and c have been used to figure out the behaviour of the physically relevantquantities graphically within the stellar interior.(ii) Fig. 1 shows that the metric potentials are positive within thestellar interior as per the requirement. Figures 3 and 4 show the vari-ations of the energy density ρ , radial pressure p r and tangential pres-sure p t , respectively. The pressures are radially decreasing outwardsfrom its maximum value at the centre and in case of radial pressureit drops to zero at the boundary as it should be but the tangentialpressure remains non-zero at the boundary. Obviously, all the quan-tities decrease monotonically from the centre towards the boundary.Variation of anisotropy has been shown in Fig. 5 which is zero at thecentre as expected and is maximum at the surface.(iii) The mass function is shown graphically in Fig. 6. Note thatthe mass function is regular at the center.(iv) One important criterion of satisfaction of energy conditions isshown graphically in Fig. 7. The matching of the interior space timewith that of the exterior metric is shown in Fig. 8.(v) In the present model the origin of the anisotropy is consid-ered from the gravitational tidal effects which caused deformation inthe structure of the matter distribution. We therefore have calculatedLove number for six different compact stars as shown in the Table 1.Here k is approximately 0.03 to 0.1 for almost all the stars with thenumerical values of parameters α = 0.22, A = 0 . . It is interestingto note that our 2 D graphical plots resemble to those of the workby Yazadjiev, Doneva & Kokkotas (2018) in connection to the tidalLove numbers of neutron stars in f ( R ) gravity. It is also to note thatour computed values are nearer to the values reported elsewhere indefferent context which ranges from 0.111 to 0.207 (Hinderer 2008;Kramm et al. 2011).However, one interesting point has been highlightedby Kramm et al. (2011) that an observational k would implya maximum possible core mass and metallicity. In connection tothe Neptune-sized exoplanet GJ b especially for k < . , itwould not help to further constrain interior models because in thatregime the solutions are too degenerate. However, a k > . MNRAS , ?? – ?? (2021) Das, Ray, Khlopov, Nandi & Parida would indicate a maximum core mass M c < . M p and largeouter envelope metallicities. ACKNOWLEDGEMENTS
SD gratefully acknowledges support from the Inter-University Cen-tre for Astronomy and Astrophysics (IUCAA), Pune, India under itsVisiting Research Associateship Programme. Research of MK wassupported by the Ministry of Science and Higher Education of theRussian Federation under Project ”Fundamental problems of cosmicrays and dark matter”, No. 0723-2020-0040.
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